For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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0
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2answers
14 views

Given x coordinate difference find the angular difference

A point on a circle moved horizontally by $x$. How to find $\alpha$ in the picture below, knowing $x$, circle radius and center? I'm pretty sure this is doable but I just can't reach the solution. ...
6
votes
3answers
107 views

Reflection relating two subspaces

Let $S_1, S_2 \subseteq \mathbb{R}^n$ be two linear $k$-dimensional subspaces. Does there always exist a hyperplane $H$ such that $S_1 = R_H S_2$, where $R_H$ denotes the orthogonal reflection across $...
0
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0answers
22 views

Clarification Needed:Equation Of Refracted Ray/Line

A ray of light is sent along the line $2x-3y=5$.After refracting across the line $x+y=1$ it enters the opposite side after turning by $15^0$ away from the line $x+y=1$.Find the equation of the ...
0
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1answer
55 views

Determine the varying area of the shaded region

How to estimate the area of the shaded region shown in the attached Figure? Note that in the figure, $p$ has a maximum and minimum values of $p_a$ and $p_b$ respectively. Moreover, $p$ follows a ...
1
vote
1answer
39 views

A rectangle of a given aspect ratio inscribed in a hexagon.

I'm trying to find the largest rectangle of a given aspect ratio that can be inscribed in a hexagon. I'm able to sort of walk through the problem in reverse, i.e. given an x, I can calculate the ...
4
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0answers
47 views

On the relationship between $\text{SL}_2(5)$ and $A_5$ [duplicate]

I have two questions. What is the quickest way to see from scratch that $\text{SL}_2(5)/\{\pm I\}$ is isomorphic to the alternating group $A_5$? Does $\text{SL}_2(5)$ have any subgroups isomorphic ...
8
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0answers
56 views

Finite subgroup of $\text{SO}(3)$ acts on set of points on unit sphere in $\mathbb{R}^3$ which are fixed via some nontrivial rotation in $G$

Let $G$ be a finite nontrivial subgroup of $\text{SO}(3)$. Let $X$ be the set of points on the unit sphere in $\mathbb{R}^3$ which are fixed by some nontrivial rotation in $G$. I have two questions. ...
1
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1answer
23 views

Find the ratio of diagonals in Trapezoid

Given $ABCD$ a rectangular trapezoid, $\angle A=90^\circ$, $AB\parallel DC$, $2AB = CD$ and $AC \perp BD$. What is the value of $AC/BD$ ? Attempts so far: I have tried using the ratio of the areas ...
0
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0answers
13 views

efficiency of different whole-number-mass-to-a-power in balancing a regular triangle/tetrahedron

I saw this qustion: http://puzzling.stackexchange.com/questions/186/whats-the-fewest-weights-you-need-to-balance-any-weight-from-1-to-40-pounds Suppose you want to create a set of weights so ...
3
votes
1answer
32 views

Geometric Shapes that can be placed inside itself

My questions title may need to be improved, and I am highly open for recommendations. Also if this is the incorrect community to post in, I would be happy to be directed to the correct one. I am ...
3
votes
3answers
31 views

Tetrahedron: Signed distance between circumcenter and face

In a triangle, the signed distance between the edge $e_1$ and the circumcenter of the triangle can be written as $$ d_1 = \frac{\langle e_2, e_3\rangle}{\langle e_1\times e_2, e_1\times e_3\rangle}...
3
votes
1answer
26 views

Determining Locations of Circles to Optimally Cover a Polygon

I want to completely cover a region on a map(Continental US)/polygon with circles of a certain radius. Is there a way to determine the best locations and how many circles would be needed to completely ...
4
votes
11answers
1k views

How to determine if two lines are parallel/ almost parallel?

I have $2$ lines in this form Line $1$: $(x_1,y_1)$ $(x_2,y_2)$ Line $2$: $(x_3,y_3)$ $(x_4,y_4)$ I want to detect if the two lines are parallel or almost parallel. My idea is to if the angle ...
27
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2answers
3k views

Name of this famous question?

I think that this question is well known but I cannot remember its name, and now I am interested in it and wanted to look it up, but cannot find anything just based on a description. If anyone knows ...
1
vote
1answer
32 views

What's the area of the triangle if one side and tan is given?

The triangle ABC is right-angled with a right angle at the corner C. Calculate the area of the triangle if c = |AB| = 10 and tan for B is 1/5. I take it I have to use the area formula with sin, but I ...
0
votes
1answer
31 views

Triangle Inequalities in Right Angled triangle.

In $\triangle{ABC}$, $\angle{ABC}=90^{\circ}$, $AB=BC$ and $AC=\sqrt{3}-1$. Suppose there exist a point $P_0$ in the plane of $\triangle{ABC}$ such that $AP_0+BP_0+CP_0 \leq AP+BP+CP$ for all points $...
0
votes
1answer
15 views

Three points and translation of the second

First of all, thanks for reading me and sorry for english mistakes. I have a programming--mathematical problem. Picture of the problem I have 3 coordonates in a 2d space, of 3 points. I want to ...
2
votes
2answers
62 views

Is there a space filling “tube”?

I am trying to construct a curve that fills the space as much as possible while maintaining certain minimum distance between two segments of the curve. Intuitively, this should be some sort of space ...
1
vote
1answer
32 views

The coincidence orthocenters of the two triangles

Let $CH -$ height in acute-angled triangle $ABC$. Some points $K$ and $N$ are on side $AB$. Let $O_1 -$ orthocenter of triangle $ACN$ and $O_2 -$ orthocenter of triangle $BCK$. Prove $$O_1=O_2=O \...
0
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0answers
35 views

Hippocrates trapezoid lune

How can I prove that a lune based on the construction of a constructible isosceles is quadrable? Hippocrates' other squarable lune
0
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0answers
15 views

Reading a 3d graph to generate a 2d projection.

I know this will sound very dum but I have spent some good time trying to understand before posting this question. Basically, I need some help in understanding how (a) and (b) are being used to ...
0
votes
0answers
19 views

Calculating fov angle based on distance

I'm trying to calculate the angle between me and the target angle yaw in a 3D game, so that the actual angle is always the same based on distance how far I am from the target. I've tried a few ...
0
votes
1answer
41 views

number of rectangles in two superimposed grids

I got two grids consisting of square "pixels", each has a different unit length per pixel though, 1 and $\frac1\xi$. Now I superimpose them as in the following image. The grid sizes differ, as ...
0
votes
2answers
47 views

Grid with both squares and equilateral triangles

Is it possible to have a grid that contains both squares and equilateral triangles? By grid I mean any set of the form $M \mathbb Z^2$, with $M \in GL_2\mathbb R$. I think this is impossible, ...
0
votes
1answer
11 views

Transitive parallel lines in noneuclidean-geometry

Is it true in neutral geometry that "If a line $m$ parallel to to a line $\ell$ , and line $\ell$ parallel to line $n$ then $m$ parallel to line $n$"? ' where $m\ne n$ I think that this is corrent, ...
0
votes
1answer
21 views

Finding the gradient vector of a plane along the plane's surface

How do you find the gradient vector of a plane? I have a plane that passes through the origin with the equation P: 5x + 95y + 46z = 0 whose normal ...
2
votes
6answers
82 views

Checking nature of angles of a triangle given the equations of the three lines that form a triangle

Suppose we have three lines $\ell_i=a_ix+b_iy=c_i$, $i=1,2,3$ and we are given that they form a triangle. I need to find which angles are acute and which are obtuse without plotting the lines ...
2
votes
2answers
42 views

Radius of inner circles given radius of outer circle and number of inner circles in circular fractal

I am trying to create a circular fractal in which each circle is composed by a given number $n$ of smaller circles. It would look something like this for $n = 8$: However, I don't know how to ...
0
votes
1answer
37 views

Finding the overlap between direction of distance in position space and direction of distance in velocity space

There are two objects A and B that can be described in position space and velocity space. The position space describes the instantaneous positions of the objects while the velocity space describes ...
0
votes
3answers
51 views

Finding ratio of cevian lines

I am preparing for an exam and doing some pratice problems. So I'm having a difficult time with this problem. At first I thought the ratio was 2:1 and then I also thought I would be able to use the ...
0
votes
0answers
20 views

Solve linear system with $A_{i,j} = \langle e_i, e_j\rangle^2$, edges of a tetrahedron

I have six vectors in $e_i\in\mathbb{R}^3$ that are the edges of a tetrahedron. Let us consider now the linear equation system $Ax=b$ with $$ A_{i,j} = \langle e_i, e_j\rangle^2,\\ b_i = \langle e_i, ...
3
votes
1answer
1k views

IMO 2016 Problem 3

Let $P = A_1 A_2 \cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $...
0
votes
1answer
74 views

Can you Prove or Disprove this?

In $a^n+b^n=c^n$ , $(a<b<c)$ , $a,b,n$ belongs to natural numbers, If $n>=b/2$ , $c$ lies between $(b,b+1)$. Also, only for $n=1,2$ , $c=b+1$.
0
votes
1answer
18 views

In flux vs. radius equation, isn't there a mistake in the dimensions?

The video: https://www.youtube.com/watch?v=m13kKLHhN6Y The equations given are: $\frac{Energy}{Seconds} = \frac{L . A}{4 \pi r^2}$ where $L$ is the luminosity, $A$ area and $r$ is the radius. $Flux ...
0
votes
2answers
55 views

How to express an angle of 90 degrees between two lines?

If I would extend two lines $l_1$ and $l_2$ they would intersect with an angle of 90 degrees. How should I write with math terms that there would be a 90 degree angle. I assume $l_1 \perp l_2$ is ...
0
votes
0answers
24 views

Explicitly calculating the group of global isometries of a convex regular polygon (dihedral group)

Instead of an intuitive geometric description of the dihedral groups $D_{2n}$, that one can find in virtually every good book on group theory, I want to calculate the global isometries of a convex ...
0
votes
1answer
30 views

Parametric equations of an ellipse in an arbitrary plane at an arbitrary orientation?

I have searched both on here and on stackoverflow for answers to this question and I can't seem to find a good answer relating to what I'm doing. I have a center point and two vectors, one that is in ...
0
votes
1answer
27 views

If we increase the radius of a circle, does the arc's length equal the length of two end points of the arc?

I'm taking an online course and in it, the professor says that if we increase the radius of a circle, the arc's length will be equal to the length of line joining the end points of the arc (https://...
-4
votes
2answers
83 views

Geometric Implication of 0.9999… =1

I have been taught that $0.\overline9=1$. Now if we think of the finite series' in turn, each closer in turn to $1$ than the previous: 0.9 0.99 0.999 etc. If we called the difference between ...
2
votes
0answers
45 views

Could Euclid have proven Dedekind's definition of real number multiplication?

In Euclid's day, the modern notion of real number did not exist; Euclid did not believe that the length of a line segment was a quantity measurable by number. But he did think it made sense to talk ...
0
votes
1answer
22 views

Pair of straight lines problem: Prove that $g (a_1+b_1)=g_1 (a+b) $

If the lines joining the origin and the point of intersection of the curves $ax^2+2hxy+by^2+2gx=0$ and $a_1x^2+2h_1xy+b_1y^2+2g_1x=0$ are mutually perpendicular then prove that $g (a_1+b_1)=g_1 (a+b) $...
2
votes
1answer
34 views

Identifying a triangle in the 3d-space as acute, obtuse, right or equilateral

Triangle $ABC$ has vertices $A(-1, 1, 3)$, $B(-1, 3, 5)$, and $C(-3, 3, 3)$. What kind of triangle is $ABC$? Justify your answer. So far all I have done is I found the distance between $AB$, $BC$ ...
1
vote
1answer
36 views

Show that the reflection of a disc through the origin is a disc

This is a problem from the book "Basic Mathematics" by S.Lang (p.225, exercise 13b). It is similar to the one in my previous question, with the exception that we're considering a reflection instead of ...
-8
votes
1answer
23 views

Find the Perimeter [closed]

Determine to the nearest tenth the perimeter of the triangle with the coordinates A(-9,6) B(-3,10) C(-2,2) Answer choice A 24.9 B 23.3 C 29.7 D 28.5
2
votes
1answer
26 views

How can I get the angle on radian on wolfram?

Here, the argument of a complex number in degrees is given. How can I ask wolfram to convert it into radians (they give an approximation).
2
votes
3answers
73 views

Motivation for the dot product

We can motivate the cross product by considering a 3D vector perpendicular to two others. This results in 3 equations in 2 unknowns, i.e. a line of solutions, and... $\lambda(u_2 v_3 - v_2 u_3, ...
2
votes
1answer
38 views

Sierpinski triangle formula: How to take into account for 0th power?

The formula to count Sierpinski triangle is 3^k-1 .It is good if you don't take the event when k=0.But how can you write a more ...
4
votes
1answer
69 views

Interesting circles hidden in Poncelet's porism configuration

This question is an investigation starting here, with a straightedge and compass construction of $ABC$ given $(R,r,h_A)$. The key lemma is the following one: Let $\Gamma$ be a circle with centre ...
1
vote
0answers
14 views

Show that the disc of radius $r$ centered at $A$ is the translation by $A$ of the disc of radius $r$ centered at the origin

This is a problem from the "Basic Mathematics" book by S.Lang (p. 225, exercise 12). My problem: Let $D(r, A)$ denote the disc of radius $r$ centered at $A$. Show that $D(r, A)$ is the translation ...
1
vote
0answers
44 views

Finding an isometry in $\mathbb R^4$ under some conditions.

This is a follow-up to a previous question of mine that was not fully answered, I tried again my hand at the problem and I think I am close to a solution. In an affine space with the standard ...